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Unit i trigonometry satyabama univesity

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Unit i trigonometry satyabama univesity

  1. 1. UNIT I TRIGONOMETRY 10 hrs. Review of Complex numbers and De Moivre’s Theorem. Expansions of Sinnθ and Cosnθ; Sinθ and Cosθ in powers of θ, Sinnθ and Cosnθ in terms of multiples of θ. Hyperbolic functions – Inverse hyperbolic functions. Separation into real and imaginary parts of complex functions I.Separate into real and imaginary parts of cos(x+iy) Find the real part of sin(x + iy). Separate into real and imaginary parts of cos(x+iy) Find the real part of Sin h (A+iB) Separate into real and imaginary parts of tan(x + i y) Separate sin (x + iy) into real and imaginary parts Separate into real and imaginary parts of cos(x+iy) Separate real and imaginary parts of cosech (x + iy). II. Write down the expansion for tan nθ interms of power of tanθ. Write down the expansion for tan nθ interms of power of tanθ. Write the expansion of sin nθ. Write the expansion of sin nθ. Write the expansion of sin nθ. Write down the expansion for tan nθ interms of power of tanθ. Write the expansion of sin nθ. Expand cos4θ in terms of cosθ. Expand sin5θ in terms of sinθ. Expand cos4θ in terms of cosθ. Expand cos θ in powers of cos θ and sin θ. Write cos4θ in terms of a series of cosines of multiples of θ Expand Cos4 θ in a series of cosines of multiples of θ. Express θ θ sin 4sin in terms of cosθ
  2. 2. If sin (A + iB) = x + iy, prove that 1 cossin 21 2 2 2 =− A y A x If cos (α + iβ) = cosθ + isinθ prove that Sin2 α = ± sinθ. 4. If x = cosθ + i sinθ, what is n x x       − 1 Show that cos4θ = 8cos4 θ - 8cos2 θ + 1 Show that 3cos4 cos 3cos 2 −= θ θ θ Show that = 4 cos2 θ – 3. Show that 3cos4 cos 3cos 2 −= θ θ θ Show that . 1 1 log 2 1 )(tanh 1       − + =− x x x Prove that tan h-1 = log x for x>0. Prove that tan h-1 = log x for x>0. Prove that cosh2 x – sinh2 x = 1. Prove that tan h-1 = log x for x>0. Show that sinh 2x = 2sinhx coshx. Prove that cosh2 x – sinh2 x = 1. PART-B 1.Separate real and imaginary parts of cosech (x + iy) (b) Separate into real and imaginary parts of tanh(x + iy).
  3. 3. (b) Separate tan-1 (x + iy) into real and imaginary parts (b) Separate tanh-1 (x + iy) into real and imaginary parts. (a) Separate into real and imaginary part of tan-1 (x + iy). (b) Separate tanh-1 (x + iy) into real and imaginary parts. (b) Separate into real and imaginary parts of tanh(x + iy). (b) Separate tan-1 (x + iy) into real and imaginary parts. (b) Separate tanh-1 (x + iy) into real and imaginary parts. II.(a) Expand sin 6θ in terms of sin θ. Find θ θ cos 7cos in terms of cosines powers of θ. (a) Expand sin 7θ as a polynomial in sin θ, Hence show that Sin π/7 sin 2π/7 sin 3π/7 sin 4π/7 sin 5π/7 sin6π/7 = -7/64 (a) Expand sin 7θ as a polynomial in sin θ, Hence show that Sin π/7 sin 2π/7 sin 3π/7 sin 4π/7 sin 5π/7 sin6π/7 = -7/64 (a) Expand sin 6θ in terms of sin θ. (a) Expand sin 6θ in terms of sinθ Expand Sin8 θ in a series of cosines of multiple of θ. . Expand Sin4 θCos3 θ in a series of cosines of multiples of θ. (may-2012) Expand Sin4 θCos3 θ in a series of cosines of multiples of θ. (a) Obtain the expansion of Sin7θ/Sinθ (a) Expand Sin3 θ. Cos5 θ in a series of sines of multiples of θ. (a) Expand sin 5 θ cos 4 θ in a series of sines of multiples of θ . (may- 2013) (a) Prove that 64sin4 θ cos3 θ = cos7θ - cos 5θ = 3 cos 3θ + 3cosθ.
  4. 4. (a) Prove that cos7θ secθ = 64cos6 θ - 112cos4 θ + 56 cos2 θ - 7. (a) Expand Sin3 θ. Cos5 θ in a series of sines of multiples of θ. It cos(u+iv) = x+iy where u,v,x,y as real, prove that (i) (1+x)2 + y2 = (Coshv + cos u)2 (ii) (1-x)2 + y2 = (Coshv – Cos u)2 (a) Prove that cos6 θ = [cos6θ + 6cos4θ+15cos2θ+10]. (a) Prove that sin6 θ = ]102cos154cos66[cos 32 1 −+−− θθθ (b) Prove that sin5 θ cos2 θ= 1/26 [sin7θ – 3sin5θ + sin3θ +5sinθ] (a) Prove that 75611264 7 246 −+−= θθθ θ θ CosCosCos Cos Cos (a) Find θ θ cos 7cos in powers of cosθ. (may-2012) (a) Find θ θ cos 7cos in powers of cosθ. Show that 3cos4 cos 3cos 2 −= θ θ θ (a) Show that [Cos 9θ + cos 7θ - 4cos 5θ - 4cos 3θ + 6cosθ] (a) Show that [Cos 9θ + cos 7θ - 4cos 5θ - 4cos 3θ + 6cosθ] (a) Show that [Cos 9θ + cos 7θ - 4cos 5θ - 4cos 3θ + 6cosθ]
  5. 5. x 2 x 2 Find θ θ cos 7cos in terms of cosines powers of θ (a) Find θ θ cos 7cos in powers of cosθ. (a) Prove that 75611264 7 246 −+−= θθθ θ θ CosCosCos Cos Cos (a) Prove that θθθ θ θ cos6cos32cos32 6 35 +−= Sin Sin (may-2012) (b) If Sin ( αθααφθ sincos,sincos) 2 ±=+=+ thatproveii If x + iy = sin (A+iB) prove that 1 cossin 1 sinhcosh 2 2 2 2 2 2 2 2 =−=++ A y A x and B y B x (b) If sin (α + iβ) = x + iy, prove that 1 sinhcosh 2 2 2 2 =+ ββ yx If cos (α + iβ) = cosθ + isinθ prove that Sin2 α = ± sinθ. (8 marks) (b) If tan x/2 = tan h y/2, prove that sin hy = tanx and y = log tan (b) If tan = tan h prove that cos x cos hx = 1. 12. It cos(u+iv) = x+iy where u,v,x,y as real, prove that (i) (1+x)2 + y2 = (Coshv + cos u)2 (may-2012) (ii) (1-x)2 + y2 = (Coshv – Cos u)2 12. It cos(u+iv) = x+iy where u,v,x,y as real, prove that (i) (1+x)2 + y2 = (Coshv + cos u)2 (ii) (1-x)2 + y2 = (Coshv – Cos u)2
  6. 6. Show that sinh 2x = 2sinhx coshx. Prove that cosh2 x – sinh2 x = 1. (b) If tan (θ+ iφ) = tanα + i secα, Prove that . 2 2 2 cot2 α π πθ αϕ ++=      ±= nande . (b) If tan x/2 = tan h y/2, prove that sin hy = tanx and y = log tan 12. (b) Show that x x xx tanh1 tanh1 2sinh2cosh − + =+ (b) If x+iy = cos(A – iB), find the value of X2 + (b) Show that tanh1 tanh1 2sinh2cosh − + =+ x xx (or) 12. (a) If , 2166 2165sin = θ θ show that θ is nearly equal to 3 1° ’ (b) If cos hu = secθ, prove that u=log tan       + 24 θπ (b) If sin( A + i B) = x + i y , prove that X2 /Sin2 A –x2 /cos2 A = 1 (b) Prove that tanh– 1 (sin θ) = cosh-1 (sec θ). Cosh2 B sinh2 B
  7. 7. x 2 x 2 (b) If sin θ = tanh x prove that tan θ = sinh x. (b) If tan = tan h prove that cos x cos hx = 1. 12. (b) If tan       2 x = tanh       2 y , prove that y = log tan       + 24 xπ Expand Sin4 θCos3 θ in a series of cosines of multiples of θ. (b) If tan (θ+ iφ) = tanα + i secα, Prove that . 2 2 2 cot2 α π πθ αϕ ++=      ±= nande (a) Expand sin 7θ as a polynomial in sin θ, Hence show that Sin π/7 sin 2π/7 sin 3π/7 sin 4π/7 sin 5π/7 sin6π/7 = -7/64 (b) If tan x/2 = tan h y/2, prove that sin hy = tanx and y = log tan 12. (b) Show that x x xx tanh1 tanh1 2sinh2cosh − + =+ UNIT II Characteristic equation of a square matrix - Eigen values and Eigen vectors of a real matrix- properties of Eigen values and Eigen vectors, Cayley-Hamilton theorem (without proof) verification – Finding inverse and power of a matrix. Diagonalisation of a matrix using similarity transformation (concept only) , Orthogonal transformation – Reduction of quadratic form to canonical form by orthogonal transformation 1.Find the rank of the matrix             − − − 2642 3963 1321 2.Find the sum and product of the eigen values of matrix.
  8. 8. 1 a1 1 a2 1 an           321 221 111 3.Find the rank of the matrix 1 3 2 4 1 4 3 2 2 7 5 6 4 14 10 12 4.If a1, a2, … an are the eigen values of a square matrix A, prove that a.For what values of a and b the equations. , , … are the eigen values of A–1 5.The product of two eigen values of the matrix           − −− − = 312 132 226 A is 16. Find the third eigen value. 6. Write down the matrix of the quadratic form x2 + y2 + z2 + xy + yz + zx. 7.Find the sum and product of the eigen values of the matrix           201 020 102 8. Prove the matrix       = 10 01 M is orthogonal. 9.State any two properties of eigen values of a matrix. 10. Use Cayley-Hamilton theorem to find the inverse of the matrix       = 62 37 A 11.Find the sum of the squares of the eigen values of A =           500 620 413 . 12.. Determine the nature of the Quadrative form without reducing to the canonical form: x2 +3y2 +6z2 +2xy+2yz+4xz. Find the eigen value and eigen vectors of
  9. 9. 6 –2 2 –2 3 –1 2 –1 3 13. Find the eigen value and eigen vectors of 2 -1 -8 4 14. The eigen values of the matrix A = the third eigen value and the product of eigen values. 15.State cayly-hamilton theorem. 16..If λ = 3 and λ = -2 are twp eigen values of           = 113 151 311 A then find third eigen value 17.Find the rank of the matrix             − − − 2642 3963 1321 18.. Find the sum and product of the eigen values of matrix.           321 221 111 19.State any two properties of eigen values of a matrix. 20.Use Cayley-Hamilton theorem to find the inverse of the matrix       = 62 37 A 21.If A=           − 300 720 321 , find the eigen values of A-1 and A 3 .M12 22. Find the nature of the quadratic form 222 32 zyx +− M12 23. Find the sum and product of the eigen values of the matrix           − − − 111 111 111
  10. 10. 24. State Cayley Hamilton theorem. 25.Find the rank of matrix.           − − − 821 712 643 26.. Find the sum and product of eigen values of the matrix           − − − 312 421 441 27.State Cayley-Hamilton theorem.-D11 28.Find the quadratic form corresponding to the matrix          − 305 002 521 D11 27.State cayly-hamilton theorem.-D11 28.If λ = 3 and λ = -2 are twp eigen values of           = 113 151 311 A then find third eigen value. D11 29.If A =       23 14 , then find the eigen values of A2 .-M11 30.Write the matrix of the quadratic form.4x2 + 2y2 – 3z2 + 2xy + 4zx –M11 31. Define rank of a matrix.-M11 32. Two eigen values of 2 2 1 33. A = 1 3 1 are equal to 1 each. Find the third eigen value.-M11 1 2 2 34.In the rank of A =           − k53 241 112 is 2, find the value of k.-D10 35. Find the sum of the squares of eigenvalues of the matrix –D10 A =           526 048 003 36.Find sum and product of Eigen values of           −−− − = 312 301 221 A .-M10 37..Write the matrix of quadratic form (x1 2 +3x2 2 +6x3 2 -2x1x2+6x1x3+5x2x3).-M10
  11. 11. 38.State any one property of Eigen value of a matrix and verify it on the matrix       23 11 .D09 39. Write down the quadratic form whose corresponding matrix –is           − −− − 623 241 312 . D09 a.1) x + y + z = 6 x + 2y + 3z = 10 x + 2y + az = b have (i) No solution (ii) A unique solution (iii) Infinite number of solutions. (or) A1. Reduce quadratic form 323121 2 3 2 2 2 1 2625 xxxxxxxxx +++++ to a canonical form through an orthogonal transformation. a.2)If A and B are any two non-singular matrices of the same order. Prove that (AB)–1 = B–1 A–1 . (or)
  12. 12. A2. If A is any square matrix, prove that ½ (A + AT ) is a symmetric matrix and ½ (A – AT ) is a skew-symmetric matrix. (a3) Show that the equations 3x + y + 2z = 3, 2x – 3y – z = -3, x + 2y + z = 4 are consistent and solve them. (b) Find the eigen values and eigen vectors of the matrix.           −− 327 112 022 (or) A3. Reduce the quadratic form 8x2 + 7y2 + 3z2 – 12xy – 8zy + 4xz to the canonical form through an orthogonal transformation. a.4)Reduce the quadratic form 323121 2 3 2 2 2 1 8412378 xxxxxxxxx −+−++ in to its canonical form by using orthogonal reduction. (or) A4. Verify Cayley – Hamilton theorem for the matrix           = 121 324 731 A Also find A– 1 and A4 . (a5) Find the Eigen values and Eigen vectors of the matrix           − = 322 121 101 A (b) Diagonalise the matrix A given above by similarity transformation. (or) A5. (a) Find the inverse of the matrix           − −= 312 321 111 A by using Cay;ey-Hamilton theorem. (b) Obtain an orthogonal transformation, which will transform the quadratic form 6x2 + 3y2 + 3z2 – 4xy – 2yz + 4zx into a canonical form. a.6) Reduce the quadratic form 2x2 + 6y2 + 2z2 + 8xz to canonical form by orthogonal reduction. Find also the nature of the quadratic form. (or) A6. (a) Find the eigen values and eigen vectors of the matrix       21 45 (b) Verify Cayley Hamilton for the marix A =           211 010 112 a.7)Find the eigen values and eigen vectors of 2 2 0 2 1 1 –7 2 –3
  13. 13. –1 2 3 8 1 –7 –3 0 8 A7. Using cayley-Hamilton theorem, find the inverse of the matrix A = a.8)Show that the quadratic form 133221 2 3 2 2 2 1 4812378 xxxxxxxxxQ +=−++= is positive semi definite. (or) A8. Investigate for what values of a and b the simultaneous equations x + y + z = 6, x + 2y + 3z = 10, x + 2y + az = b. will have (a) no solution (b) unique solution (c) infinite solution a.9)For what values of a and b the equations. x + y + z = 6 x + 2y + 3z = 10 x + 2y + az = b have (i) No solution (ii) A unique solution (iii) Infinite number of solutions. (or) A9. Reduce quadratic form 323121 2 3 2 2 2 1 2625 xxxxxxxxx +++++ to a canonical form through an orthogonal transformation. (a.10) Find the Eigen values and Eigen vectors of the matrix           − = 322 121 101 A (b) Diagonalise the matrix A given above by similarity transformation. (or) A10. (a) Find the inverse of the matrix           − −= 312 321 111 A by using Cay;ey-Hamilton theorem. (b) Obtain an orthogonal transformation, which will transform the quadratic form 6x2 + 3y2 + 3z2 – 4xy – 2yz + 4zx into a canonical form.
  14. 14. a.11)Verify Cayley – Hamilton theorem for the matrix A=           122 212 221 and hence find A-1 and A4 M-12 (or) A11. Reduce the quadratic form 3x yzxzxyzy 22235 222 −+−++ into a canonical form by orthogonal reduction.-M12 a.12). Diagonalize the matrix           113 151 311 by orthogonal transformation. (or) A12. (a) Show that the matrix           − −= 111 112 301 A satisfies its own characteristic equation and hence find A-1 .-M12 (b) Find the eigen values and eigen vectors of the matrix           110 110 001 M12 a.13)State Cayley–Hamilton theorem and find the inverse of the matrix A =           − 200 422 201 using Cayley – Hamilton theorem hence find A4 . (or) A13. Reduce 6x2 + 3y2 + 3z2 – 4xy – 2yz + 4xz into canonical form by an orthogonal transformation a.14) Reduce the quadratic form 133221 2 3 2 2 2 1 4812378 xxxxxxxxx +−−++ into its canonical form using orthogonal reduction.-D11 (or) A14.Using Cayley-Hamilton theorem find the inverse of the matrix           − − − = 803 718 301 A D11 a.15)Show that the quadratic form 133221 2 3 2 2 2 1 4812378 xxxxxxxxxQ +=−++= is positive semi definite. (or)
  15. 15. A15. Investigate for what values of a and b the simultaneous equations x + y + z = 6, x + 2y + 3z = 10, x + 2y + az = b. will have (a) no solution (b) unique solution (c) infinite solution a.16). Find the eigen values and eigen vectors of (a) the matrix           − −− − 342 476 268 (b) Verify Cayley-Hamilton theorem for the matrix A =           − − 111 112 301 . Hence find its inverse. (or) A16. Reduce the quadratic form 3x1 2 +5x2 2 +3x3 2 – 2x2x3 + 2x3x1 – 2x1x2 to a canonical form by orthogonal reduction. Find also index, signature and nature of the quadratic form. (a17) Verify Cayley-Hamilton theorem for the matrix =M11 7 2 –2 A = –6 –1 2 6 2 –1 2 2 –7 (b) Find the eigen values and eigen vectors of 2 1 2 0 1 -3 (or) A17. Reduce 6x2 + 3y2 – 4xy – 2yz + 4xz + 3z2 into a canonical form by an orthogonal reduction. Discuss the nature of quadratic form.-M11 a.18)Using cayley.Hamilton theorem find A-1 if           − − − = 573 452 221 A ; Also verify the theorem.-D10 (or) A18. Reduce the equation form 10x2 + 2y2 + 5z2 + 6yz – 10zx – 4xy to a canonical form.D10 a.19). Verify Cayley-Hamilton theorem for the matrix           − −− − = 211 121 212 A . Hence compute A- 1 . –M10 (or) A19. Reduce the matrix           = 204 060 402 A to diagonal form by orthogonal transformation-M10
  16. 16. (a.20) Find the Eigen values and Eigen vectors of the matrix           − − = 310 212 722 A (b) Diagonalise the matrix       = 23 14 A hence find A8 .-D09 (or) A20. (a) Find the inverse of the matrix           −− − = 126 216 227 A using Cayley-Hamilton Therorem. –D09 UNIT III GEOMETRICAL APPLICATIONS OF DIFFERENTIAL CALCULUS Curvature –centre, radius and circle of curvature in Cartesian co-ordinates only – Involutes and evolutes – envelope of family of curves with one and two parameters – properties of envelopes and evolutes – evolutes as envelope of normal. UNIT IV FUNCTIONS OF SEVERAL VARIABLES 1 Functions of two variables – partial derivatives – Euler’s theorem and problems - Total differential – Taylor’s expansion – Maxima and minima – Constrained maxima and minima – Lagrange’s multiplier method – Jacobian – Differentiation under integral sign. UNIT V ORDINARY DIFFERENTIAL EQUATION Second order linear differential equation with constant coefficients – Particular Integrals for eax, sin ax, cos ax, xn, xneax, eax sinbx, eax cos bx. Equations reducible to Linear equations with constant co-efficient using x=et. Simultaneous first order linear equations with constant coefficients - Method of Variations of Parameters.

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